Spatiality of derivations on the algebra of $\tau$-compact operators

This paper is devoted to derivations on the algebra $S_0(M,\tau )$ of all $\tau$-compact operators affiliated with a von Neumann algebra $M$ and a faithful normal semi-finite trace $\tau \mathrm{.}$ The main result asserts that every $t_{\tau }$-continuous derivation $D:S_0(M,\tau )\to S_0(M,\tau )$ is spatial and implemented by a $\tau$-measurable operator affiliated with $M$, where $t_{\tau }$ denotes the measure topology on $S_0(M,\tau )$. We also show the automatic $t_{\tau }$-continuity of all derivations on $S_0(M,\tau )$ for properly infinite von Neumann algebras $M$. Thus in the properly infinite case the condition of $t_{\tau }$-continuity of the derivation is redundant for its spatiality.